Proofs of convergence of random variables

Proofs of convergence of random variables mean that we are trying to figure out if a random variable will always be the same, even when it is chosen differently. To do this, we look at the numbers that the random variable produces when it is chosen over and over again. We want to see if the numbers always stay the same or if they are always different. If they are always the same, then we know that the variable is converging and will always be the same. If they are always different, then we know that the variable is not converging and will always be different.